jmc

Letting $x=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2-\cdots }}}}$, we have $x=\sqrt{2-x}$. Squaring both sides gives $x^2=2-x$, so $x^2+x-2=0$. Factoring the left side gives $(x+2)(x-1)=0$. Therefore, $x=-2$ or $x=1$. Clearly $x$ must be positive, so we have $x=\boxed{1}$.